2 Answers
2

This is not a bad question. I'd appreciate it if you expanded some to give context and explain what you're thinking about.

Short answer: No, there isn't. It's not clear that M-theory has a description in the Lagrangian framework.

Longer answer: The nature of spacetime in M-theory is radically different from the nature of spacetime in classical & quantum field theory. Exactly how different is still under investigation. But it is not clear that spacetime is infinitely divisible in this theory. If short distances don't exist, then it is not clear we should be using the Lagrangian framework to describe fundamentla physics, since these implicitly associate degrees of freedom to all distance scales.

That said: Lagrangians do play a role in the physics of M-theory. They are used to describe worldvolume QFTs, which are effective QFTs which describe how strings and branes see the classical spacetime around them: The matrix model of D0-branes. The D3-branes of AdS/CFT are described by N=4 gauge theories. The basis of perturbative string theory is a nonlinear sigma model. A stack of 5-branes in M-theory is described by a (2,0) theory.

well, the rule of thumb should be something like; allowed physical states must not create event horizons (real or virtual) so this filter the function space where path integrals should be taken, in particular it filters most of the UV degrees of freedom around or below planck scale. Why discard states that produce event horizons? because they sink quantum states into thermal distributions. The info may still be there but they should not contribute to physical amplitudes. The particular details of the function space (wordline, worldsheet or brane) are not relevant
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lurscherJun 29 '12 at 15:49

the reason that they should not contribute to physical amplitudes should be clear: they represent mixed states
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lurscherJun 29 '12 at 16:26

@lurscher: that's vaguely the idea, but I think it's bound to be misleading on the details. In particular, I'd bet the complete description of black hole radiation is unitary. But string theory is very soft in the UV, and there perhaps should be/or is a picture in which the softness is related to the difficulties signals have in escaping from event horizons.
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user1504Jun 29 '12 at 22:40

@lurscher: The problem is that whether a physical state creates an event horizon or not is not something you can determine by local measurements (there can always be more matter elsewhere making a horizon together with this matter), and when there is a fluctuating metric, it's completely hopeless. This is why strings are a miracle--- you describe them in AdS/CFT, and they solve the problem of quantum gravity without touching the issue of spacetime action or local spacetime degrees of freedom, since they are only formulated asymptotically on boundaries.
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Ron MaimonJun 30 '12 at 2:40

@lurscher: Further, the little black holes don't do magic mixing of states, they are just an unitary as an atom. It is misleading to think that micro black holes thermalize what they come in contact with, they are too small to do that.
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Ron MaimonJun 30 '12 at 2:42

There is a candidate for M-theory lagrangian for coincident M2 branes. This lagrangian may provide the first M –theory lagrangian describing the quantum dynamics of membranes. http://arxiv.org/abs/hep-th/0611108

This is a Lagrangian for M2 branes, not for "M-theory". OP wants an M-theory spacetime Lagrangian, not an AdS/CFT dual Lagrangian, and this probably doesn't exist.
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Ron MaimonJun 30 '12 at 2:38

@Ron: Yes only a dual M2 lagrangian, and also very speculative, but is a quantum membrane lagrangian. It is not a full M-theory, but when it was proposed it generate some enthusiasm in the physics community as a consistent membrane action, not full M-theory of course.
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Ernesto UlloaJul 3 '12 at 23:30

I don't understand why you still keep this answer around. This is not the question of space-time action, this would be answered by "membrane field theory" (the non-existing extension of string field theory to M-theory), and this doesn't exist, and it plausibly cannot exist. The membrane Lagrangian is important, and it reproduces a full M-theory in the same way as Matrix Theory or AdS/CFT, it's the whole shebang in certain limits.
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Ron MaimonJul 4 '12 at 5:08